2021 AMC 12B

How many integer values of $x$ satisfy $|x| \lt 3\pi?$

At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?

Suppose $$2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+x}}}=\dfrac{144}{53}.$$ What is the value of $x?$

Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84,$ and the afternoon class’s mean score is $70.$ The ratio of the number of students in the morning class to the number of students in the afternoon class is $\dfrac{3}{4}.$ What is the mean of the scores of all the students?

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y=-x.$ The image of $P$ after these two transformations is at $(-6,3).$ What is $b-a?$

An inverted cone with base radius $12$ cm and height $18$ cm is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24$ cm. What is the height in centimeters of the water in the cylinder?

Let $N=34\cdot 34\cdot 63\cdot 270.$ What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N?$

Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,$ $38,$ and $34.$ What is the distance between two adjacent parallel lines?

What is the value of the following expression? $$\dfrac{\log_2 80}{\log_{40}2}-\dfrac{\log_2 160}{\log_{20}2}$$

Two distinct numbers are selected from the set $\{1,2,3,4,\ldots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?

Triangle $ABC$ has $AB=13,$ $BC=14,$ and $AC=15.$ Let $P$ be the point on $\overline{AC}$ such that $PC=10.$ There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S,$ then the average value (arithmetic mean) of the integers remaining is $32.$ If the least integer in $S$ is also removed, then the average value of the integers remaining is $35.$ If the greatest integer is then returned to the set, the average value of the integers rises to $40.$ The greatest integer in the original set $S$ is $72$ greater than the least integer in $S.$ What is the average value of all the integers in the set $S?$

How many values of $\theta$ in the interval $0\lt\theta\le 2\pi$ satisfy the following equation? $$1-3\sin\theta+5\cos 3\theta=0$$

Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD.$ Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},$ $\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$

The figure is constructed from $11$ line segments, each of which has length $2.$ The area of pentagon $ABCDE$ can be written as $\sqrt m+\sqrt n,$ where $m$ and $n$ are positive integers. What is $m+n?$

Let $g(x)$ be a polynomial with leading coefficient $1,$ whose three roots are the reciprocals of the three roots of $f(x)=x^3+ax^2+bx+c,$ where $1\lt a\lt b\lt c.$ What is $g(1)$ in terms of $a,$ $b,$ and $c?$

Let $ABCD$ be an isosceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB\gt CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2,$ $3,$ $4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\dfrac{AB}{CD}?$

Let $z$ be a complex number satisfying $12|z|^2$ $=2|z+2|^2+|z^2+1|^2+31.$ What is the value of $z+\dfrac{6}{z}?$

Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\dfrac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\dfrac{1}{12}.$ What is the least possible number of faces on the two dice combined?

Let $Q(z)$ and $R(z)$ be the unique polynomials such that $$\begin{aligned} &z^{2021}+1 \\ &\quad = (z^2+z+1)Q(z)+R(z) \end{aligned}$$ and the degree of $R$ is less than $2.$ What is $R(z)?$

Let $S$ be the sum of all positive real numbers $x$ for which $$x^{2^{\sqrt2}}=\sqrt2^{\,2^x}.$$ Which of the following statements is true?

Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one “wall” among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),$ $\ (2,1,2),$ $\ (4),$ $\ (4,1),$ $\ (2,2),$ or $(1,1,2).$ Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?

Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,$ $2,$ $3,$ $\ldots.$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\dfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3,$ $17,$ and $10.$) What is $p+q?$

Let $ABCD$ be a parallelogram with area $15.$ Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points. Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$

Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\dfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$